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1 Chapter 1 : Mechanism design - Wikipedia The present book treats a highly specialized topic, namely effecâ tivity functions, which are a tool for describing the power structure implicit in social choice situations of various kind. Then there are lk possible aggregation rules: Thus the number of possible aggregation rules grows exponentially with the number of admissible profiles and the number of possible decision outcomes. To select an aggregation rule non-arbitrarily from this large class of possible ones, some constraints are needed. I now consider three formal arguments for majority rule. May introduced four such requirements: May proved the following: An aggregation rule satisfies universal domain, anonymity, neutrality, and positive responsiveness if and only if it is majority rule. Apart from providing an argument for majority rule based on four plausible procedural desiderata, the theorem helps us characterize other aggregation rules in terms of which desiderata they violate. Dictatorships and weighted majority rules with unequal individual weights violate anonymity. Asymmetrical supermajority rules under which a supermajority of the votes, such as two thirds or three quarters, is required for a decision in favour of one of the alternatives, while the other alternative is the default choice violate neutrality. This may sometimes be justifiable, for instance when there is a presumption in favour of one alternative, such as a presumption of innocence in a jury decision. Symmetrical supermajority rules under which neither alternative is chosen unless it is supported by a sufficiently large supermajority violate positive responsiveness. A more far-fetched example of an aggregation rule violating positive responsiveness is the inverse majority rule here the alternative rejected by a majority wins. Suppose the aim is to make a judgment on some procedure-independent fact or state of the world, denoted X. The goal is to reach a factually correct collective judgment. Under these assumptions, majority voting is a good truth-tracker: One can further show that, if the two states of the world have an equal prior probability i. Although the jury theorem is often invoked to establish the epistemic merits of democracy, its assumptions are highly idealistic. The competence assumption is not a conceptual claim but an empirical one and depends on any given decision problem. Similarly, whether the independence assumption is true depends on the decision problem in question. Finally, game-theoretic work challenges an implicit assumption of the jury theorem, namely that voters will always reveal their judgments truthfully. Even if all voters prefer a correct to an incorrect collective judgment, they may still have incentives to misrepresent their individual judgments. This can happen when, conditional on the event of being pivotal for the outcome, a voter expects a higher chance of bringing about a correct collective judgment by voting against his or her own private judgment than in line with it Austin-Smith and Banks ; Feddersen and Pesendorfer It does not require the existence of an independent fact or state of the world that the collective decision is supposed to track. Suppose each voter gets some utility from the collective decision, which depends on whether the decision matches his or her vote preference: Relatedly, majority rule minimizes the number of frustrated voters defined as voters on the losing side and maximizes total utility across voters. Brighouse and Fleurbaey generalize this result. The Rae-Taylor theorem rests on an implicit equal-stakes assumption, i. The basic model is as follows. For generality, the requirement that R be complete and transitive is not built into the definition of a preference aggregation rule. The paradigmatic example of a preference aggregation rule is pairwise majority voting, as discussed by Condorcet. As we have seen, this does not guarantee transitive social preferences. It can be shown that the proportion of preference profiles among all possible ones that lead to cyclical majority preferences increases with the number of individuals n and the number of alternatives X. However, the probability of cycles can be significantly lower under certain systematic, even small, deviations from an impartial culture List and Goodin Appendix 3; Tsetlin, Regenwetter, and Grofman ; Regenwetter et al. The domain of F is the set of all logically possible profiles of complete and transitive individual preference orderings. Independence of irrelevant alternatives: The weak Pareto principle requires that when all individuals strictly prefer alternative x to alternative y, so does society. Note that pairwise majority voting satisfies all of these conditions except ordering. It is evident that this result carries over to the aggregation of other kinds of orderings, as distinct from preference orderings, such as i belief orderings over several hypotheses ordinal credences, ii multiple criteria that a single decision maker Page 1

2 may use to generate an all-things-considered ordering of several decision options, and iii conflicting value rankings to be reconciled. Morreau forthcoming, evidence amalgamation e. The best-known cohesion condition is single-peakedness Black Single-peakedness is plausible in some democratic contexts. If the alternatives in X are different tax rates, for example, each individual may have a most preferred tax rate which will be lower for a libertarian individual than for a socialist and prefer other tax rates less as they get more distant from the ideal. Black proved that if the domain of the aggregation rule is restricted to the set of all profiles of individual preference orderings satisfying single-peakedness, majority cycles cannot occur, and the most preferred alternative of the median individual relative to the relevant left-right alignment is a Condorcet winner assuming n is odd. Other domain-restriction conditions with similar implications include single-cavedness, a geometrical mirror image of single-peakedness Inada, separability into two groups ibid. Sen showed that all these conditions imply a weaker condition, triple-wise value-restriction. There has been much discussion on whether, and under what conditions, real-world preferences fall into such a restricted domain. It has been suggested, for example, that group deliberation can induce single-peaked preferences, by leading participants to focus on a shared cognitive or ideological dimension Miller ; Knight and Johnson ; Dryzek and List Experimental evidence from deliberative opinion polls is consistent with this hypothesis List, Luskin, Fishkin, and McLean, though further empirical work is needed. An aggregation rule that produces transitive but often incomplete social preferences is the Pareto dominance procedure: An aggregation rule that produces complete but often intransitive social preferences is the Pareto extension procedure: Both rules have a unanimitarian spirit, giving each individual veto power either against the presence of a weak social preference for x over y or against its absence. Gibbard proved that even if we replace the requirement of transitivity with what he called quasi-transitivity, the resulting possibilities of aggregation are still very limited. Call a preference relation R quasi-transitive if the induced strict relation P is transitive while the indifference relation I need not be transitive. In an oligarchy, the oligarchs are jointly decisive and have individual veto power. Gibbard proved the following: One case in which we may lift it is that of spurious unanimity, where a unanimous preference for x over y is based on mutually inconsistent reasons e. Two men may each prefer to fight a duel alternative x to not fighting it alternative y because each over-estimates his chances of winning. There may exist no mutually agreeable probability assignment over possible outcomes of the duel i. In this case, the unanimous preference is a bad indicator of social preferability. This example, however, depends on the fact that the alternatives of fighting and not fighting are not fully specified outcomes but uncertain prospects. Arguably, the weak Pareto principle is more plausible in cases without uncertainty. Here we interpret the aggregation rule as a method a social planner can use to rank social alternatives in an order of social welfare. Suppose each individual in society is given some basic rights, to the effect that his or her preference is sometimes socially decisive i. Sen asked us to imagine that Lewd most prefers that Prude read the book alternative x, second-most prefers that he read the book himself alternative y, and least prefers that neither read the book z. Prude most prefers that neither read the book z, second-most prefers that he read the book himself x, and least prefers that Lewd read the book y. Assuming Lewd is decisive over the pair y and z, society should prefer y to z. Assuming Prude is decisive over the pair x and z, society should prefer z to x. So, we are faced with a social preference cycle. There exists no preference aggregation rule satisfying universal domain, acyclicity of social preferences, the weak Pareto principle, and minimal liberalism. The result suggests that if we wish to respect individual rights, we may sometimes have to sacrifice Paretian efficiency. Almost all familiar voting methods over three or more alternatives that involve some form of preferential voting with voters being asked to express full or partial preference orderings violate this condition. A standard example is plurality rule: Informally, alternatives are socially ranked in the order of how many individuals most prefer each of them. Most notably, an alternative that is majority-dispreferred to every other alternative may win under plurality rule: A second example of a preference aggregation rule that violates independence of irrelevant alternatives is the Borda count e. Informally, each voter assigns a score to each alternative, which depends on its rank in his or her preference ranking. Alternatives are then socially ordered in terms of the sums of their scores across voters: To see how this violates independence of irrelevant alternatives, consider the two profiles of individual preference orderings over four alternatives x, y, z, w in Page 2

3 Tables 3 and 4. A profile of individual preference orderings Individual 1. Page 3

4 Chapter 2 : CiteSeerX â Citation Query Effectivity Functions in Social Choice. Theory and Decision Libra Effectivity Functions in Social Choice (Theory and Decision Library C) [J. Abdou, Hans Keiding] on blog.quintoapp.com *FREE* shipping on qualifying offers. The present book treats a highly specialized topic, namely effecâ tivity functions, which are a tool for describing the power structure implicit in social choice situations of various kind. Individual preference can be modeled in terms of an economic utility function. Then the ability to create a social welfare function depends crucially on the ability to compare utility functions. This is called interpersonal utility comparison. Following Jeremy Bentham, utilitarians have argued that preferences and utility functions of individuals are interpersonally comparable and may therefore be added together to arrive at a measure of aggregate utility. Utilitarian ethics call for maximizing this aggregate. Lionel Robbins questioned whether mental states, and the utilities they reflect, can be measured and, a fortiori, interpersonal comparisons of utility as well as the social choice theory on which it is based. Consider for instance the law of diminishing marginal utility, according to which utility of an added quantity of a good decreases with the amount of the good that is already in possession of the individual. It has been used to defend transfers of wealth from the "rich" to the "poor" on the premise that the former do not derive as much utility as the latter from an extra unit of income. Robbins, pp. Apologists of the interpersonal comparison of utility have argued that Robbins claimed too much. John Harsanyi agrees that full comparability of mental states such as utility is never possible but believes, however, that human beings are able to make some interpersonal comparisons of utility because they share some common backgrounds, cultural experiences, etc. In the example from Amartya Sen, p. Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption. Sen proposes, however, that comparability of interpersonal utility need not be partial. A starving peasant may have a particularly sunny disposition and thereby derive high utility from a small income. This fact should not nullify, however, his claim to compensation or equality in the realm of social choice. Social decisions should accordingly be based on immalleable factors. Sen proposes interpersonal utility comparisons based on a wide range of data. We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. The initial results emphasized the impossibility of satisfactorily providing a social choice function free of dictatorship and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties. Page 4

5 Chapter 3 : effectivity_functions_in_social_choice_1st_edition The present book treats a highly specialized topic, namely effec- tivity functions, which are a tool for describing the power structure implicit in social choice situations of various kind. We present a modal logic for reasoning about what groups of agents can bring about by collective action. Given a set of states, we introduce game frames which associate with every state a strategic game among the agents. Game frames are essentially extensive games of perfect information with simulta Game frames are essentially extensive games of perfect information with simultaneous actions, where every action profile is associated with a new state, the outcome of the game. A coalition of players is effective for a set of states in a game if the coalition can guarantee the outcome of the game to lie in. We propose a modal logic Coalition Logic to formalize reasoning about effectivity in game frames, where expresses that coalition is effective for. An axiomatization is presented and completeness proved. Coalition Logic provides a unifying game-theoretic view of modal logic: Since nondeterministic processes and extensive games without parallel moves emerge as particular instances of game frames, normal and non-normal modal logics correspond to 1- and 2-player versions of Coalition Logic. We add a rule for Nash-consistency to Coalition Logic, a modal logic for reasoning about the abilities and rights of groups in multiagent systems. Rights of agents constitutions can be formalised using Coalition Logic, and the additional inference rule of Nash-consistency will guarantee that any m Rights of agents constitutions can be formalised using Coalition Logic, and the additional inference rule of Nash-consistency will guarantee that any multi-agent system implementing these rights will be stable, i. Show Context Citation Context Nash-Consistent Mechanisms In this section we formally dene strategic games and game forms mechanisms as is standard in game theory [6]. This result is a small adaptation of a resultsrst obtained in [10] to the class of play Effectivity functions and efficient coalitions in Boolean games. Boolean games are a logical setting for representing strategic games in a suc-cinct way, taking advantage of the expressive power and conciseness of propositional logic. A Boolean game consists of a set of players, each of which controls a set of propositional variables and has a specific A Boolean game consists of a set of players, each of which controls a set of propositional variables and has a specific goal expressed by a propositional formula. We show here that Boolean games are a very simple setting, yet sophisticated enough, for analysing the formation of coalitions. Due to the fact that players have dichotomous preferences, the following notion emerges naturally: We study the properties of efficient coalitions. In this section we study Boolean games from the point of view of effectivity functions. Our aim is to give an exact characterization of effectivity functions induc McGarvey has shown that any irreflexive and anti-symmetric relation can be obtained as a relation induced by majority rule. We address the analogous issue for dominance relations of finite cooperative games with non-transferable utility coalitional NTU games. We find any irreflexive relatio We find any irreflexive relation over a finite set can be obtained as the dominance relation of some finite coalitional NTU game. We study the structure of unstable local effectivity functions defined for n players and p alternatives. A stability index based on the notion of cycle is introduced. In the particular case of simple games, the stability index is closely related to the Nakamura Number. In general it may be any integ In general it may be any integer between 2 and p. We prove that the stability index for maximal effectivity functions and for maximal local effectivity functions is either 2 or 3. We end this section by stating the main result that justifies the introduction of cycles. For a more general result that covers the case of local effectivity functions we refer to Abdou and Keiding, Theorem 6. Effectivity functions for finitely many players and alternatives are con-sidered. It is shown that every monotonic and superadditive effectiv-ity function can be augmented with equal chance lotteries to a finite lottery modelâ i. In other words, there exists a finite game form which represents the lottery model and which has a Nash equilibrium for any profile of utility functions, where lotteries are evaluated by their expected utility. No additional condition on the original effec-tivity function is needed. A strong representation of a committee, formalized as a simple game, on a convex and closed set of alternatives is a game form with the members of the committee as players such that i the winning coalitions of the simple game are exactly those coalitions, which can get any given alternative indepe In the paper, it is Page 5

6 investigated whether committees have representations on convex and compact subsets of Rm. This is shown to be the case if there are vetoers; for committees with no vetoers the existence of strong representations depends on the structure of the alternative set as well as on that of the committee its Nakamura-number. Thus, if A is strictly convex, compact, and has smooth boundary, then no committee can have a strong representation on A. On the other hand, if A has non-smooth boundary, representations may exist depending on the Nakamura-number if it is at least 7. A representation of an effectivity function is a game form with the sa A representation of an effectivity function is a game form with the same power structure as that specified by the effectivity function. In the present paper we investigate the continuity properties of the outcome functions of such representation. It is shown that while it is not in general possible to find continuous representations, there are important subfamilies of effectivity functions for which continuous representations exist. Moreover, it is found that in the study of continuous representations one may practically restrict attention to effectivity functions on the Cantor set. Here it is found that general effectivity functions have representations with lower or upper semicontinuous outcome function. In the last decades logics for describing coalitional power in Multi Agent Systems have flourished. Roughly speaking they are all multimodal logic Nevertheless, as it happens in many real and artificial cases, things can go wrong and a desirable property may not be reached. One issue is then to find out which agent or group is responsible for such failure, in order to identify or punish it, or even remove it from the system. This is reflected in many applications, in which the environment has some interference in the course of events that will take place. In some of these Show Context Citation Context It is clear that if Agt forces an interval that comprises only values higher than the Estate, then it forces bankruptcy. Suppose now an external mechanism could interfere in the choice of Agt, by imposing its claim to cause bankruptcy. An example of such m Page 6

7 Chapter 4 : CiteSeerX â Citation Query Effectivity functions in social choice CONTENTS Preface ix 1. Introduction 1 1. Collective decisions and power structure 1 2. Social choice 3 3. Games and game forms The effectivity functions of a game form Cooperation logics have recently begun to attract attention within the multi-agent systems community. Using a cooperation logic, it is possible to represent and reason about the strategic powers of agents and coalitions of agents in game-like multi-agent systems. These powers are generally assumed t These powers are generally assumed to be implicitly defined within the structure of the environment, and their origin is rarely discussed. In this paper, we study a cooperation logic in which agents are each assumed to control a set of propositional variablesthe powers of agents and coalitions then derive from the allocation of propositions to agents. The basic modal constructs in this Coalition Logic of Propositional Control CL-PC allow us to express the fact that a group of agents can cooperate to bring about a certain state of affairs. Dunne - Artificial Intelligence, " We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices, with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games QCGs are a Such qualitative coalitional games QCGs are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining QCGs, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a QCG is non-empty is D 1 -complete. We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research. Show Context Citation Context An effectivity function E is typically defined with respect to a set of agents Ag, as with conventional coalitional games, above and a set of outcome states, S, which intuitively capture the possi We examine properties of a model of resource allocation in which several agents exchange resources in order to optimise their individual holdings. The schemes discussed relate to well-known negotiation protocols proposed in earlier work and we consider a number of alternative notions of "rat While it is known that imposing particular rationality and structural restrictions on the form of exchanges may render these unable to realise every reallocation of the resource set, in this paper we address the issue of the number of restricted rational exchanges that may be required to implement a particular reallocation when it is possible to do so. We construct examples showing that this number may be exponential in the number of resources m, even when all of the agent utility functions are monotonic. We further show that k agents may achieve in a single exchange a reallocation requiring exponentially many rational exchanges if at most k 1 agents can participate, this same reallocation being unrealisable by any sequences of rational exchanges in which at most k 2 agents are involved. Chapter 5 : Social Choice Theory (Stanford Encyclopedia of Philosophy) An effectivity function assigns to each coalition of individuals in a society a family of subsets of alternatives such that the coalition can force the outcome of society's choice to be a member of each of the subsets separately. Chapter 6 : Library Resource Finder: Location & Availability for: Effectivity functions in social choice In addition, (ii) has important applications in social choice, game, and even graph theories. Constructive proofs of (i) were given by Moulin, in, and by Peleg, in Both constructions are elegant, yet, sets of strategies Xi of players i ∈ I might be doubly exponential in size of the input EFF E. Chapter 7 : Social choice theory - Wikipedia Page 7

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